From the fact that the space \(\mathfrak {m}_{\varphi _*,w}^0\)  is rearrangement invariant Banach sequence space and has the Fatou property, we always have \((\mathfrak {m}_{\varphi _*,w}^0)_a = ((\lambda _{\varphi ,w})^{\prime })_a = ((\lambda _{\varphi ,w})^{\prime })_b = (\mathfrak {m}_{\varphi _*,w}^0)_b\)  by [1]}. If \(\varphi \)  satisfies the \(\Delta _2^0\) -condition, then \(\lambda _{\varphi , w}\)  is order-continuous [2]}. Hence by Theorem REF .(ii), the space \(\lambda _{\varphi , w}\)  has the RNP.
Conversely, if \(\lambda _{\varphi ,w}\)  has the RNP, then the space is order-continuous by Theorem REF .(i). Since \(\lambda _{\varphi ,w}\)  is order-continuous if and only if \(\varphi \)  satisfies the \(\Delta _2^0\) -condition [2]}, we obtain the desired claim.
